#94599

order	512
nilpotency class	6
isoclinism	isoclinic class of #7515
automorphism	?
homology	Abelianization (H₁) = C₂×C₈ · Schur multiplier (H₂) = C₂

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#1402
commutator	#127	C₂×C₈
Frattini, cosocle	#1962	C₂²
socle	C₂	#7375
prime radical	#532	C₂×C₄
LCS	G ▷ #127 ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1962 ◁ G
LECS	G ▷ #1962 ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

140 conjugacy classes of subgroups; only the 26 normal subgroups are shown.

normal subgroups only

This group has 140 conjugacy classes of subgroups — too many to draw. Only its 26 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 2 generators on 32 points:

(1,2,5,10,6,11,21,29,8,13,22,30,7,12,4,19)(3,14,16,26,17,31,32,9,18,24,15,25,23,28,20,27);(1,3)(2,9,11,25,13,27,12,26)(4,20)(5,16)(6,17)(7,23)(8,18)(10,28,19,24,30,31,29,14)(15,22)(21,32)

Polycyclic presentation:

2^9|a^2=d,b^2=e,d^2=f,e^2=hi,f^2=i,g^2=i,b^a=bc,c^a=ce,e^a=eg,g^a=gh,h^a=hi,c^b=cgi,d^b=dehi,g^b=gh,d^c=dgh,e^c=ehi,g^c=gi,e^d=ehi,g^d=gi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1795);